Using the first product theorem for arrays1 or the bridge theorem2, the directivity pattern of an arrangement of sources can be constructed by multiplying the directivity of one single source by a point source in the Fourier space.
A piston with radius $a$ is placed on a baffled plane
The directivity for two pistons with relatively phase $\Phi_{12}$ is \begin{equation}\large D(\theta,\phi) = D_p(\theta)\left[1 + e^{i(kd\sin\theta\cos\phi+\Phi_{12})}\right] \end{equation}
The directivity for when $\Phi_{12} =\pi$ is \begin{equation}\large D(\theta,\phi) = D_p(\theta)\left[1 - e^{ikd\sin\theta\cos\phi}\right] \end{equation} and for different values of $ka$
In this case, 2 pistons are vibrating with the same phase and the third in phase oppossition.
The directivity is \begin{equation}\large D(\theta,\phi) = D_p(\theta)\left[-1 + e^{ikd\alpha} - e^{i2kd\alpha}\right] \end{equation} where $\alpha = \sin\theta\cos\phi$. For different values of $ka$
In this case, 2 pistons are vibrating with the same phase and the third in phase-opposition.
The directivity is \begin{equation}\large D(\theta,\phi) = D_p(\theta)\left[1 - e^{ikd\alpha} - e^{ikd\beta}\right] \end{equation} where $\alpha = \sin\theta\cos\phi$ and $\beta = \sin\theta\sin\phi$. For different values of $ka$
Here, 2 pistons are vibrating with the same phase and the other two in phase opposition.
The directivity is \begin{equation}\large D(\theta,\phi) = D_p(\theta)\left[-1 + e^{ikd\alpha} - e^{ik2d\alpha} + e^{ik3d\alpha}\right] \end{equation} where $\alpha = \sin\theta\cos\phi$. For different values of $ka$
Again, two pistons are vibrating with the same phase and the other two in phase opposition.
where $\alpha = \sin\theta\cos\phi$ and $\beta = \sin\theta\sin\phi$. For different values of $ka$
[1] E. G. Williams, FOURIER ACOUSTICS Sound Radiation and Nearfield Acoustical Holography. 1999.
[2]L. L. Beranek and T. J. Mellow, Acoustics Sound Fields, Transducers and Vibration, Second edition. Academic Press, 2019. doi: 10.1016/B978-0-12-391421-7.00013-0.
Please refer to the paper : García A et al,Radiation efficiency of a distribution of baffled pistons with arbitrary phases, JASA, 2022 for more information.